1. The user provided several limit expressions and functions to analyze. 2. Let's clarify and solve each limit step-by-step. --- **Problem 1:** Evaluate $$\lim_{x \to 3} \frac{3 - \sqrt{5x + 4}}{\sqrt{x + 3} - 2}$$ Step 1: Substitute $x=3$ directly: $$3 - \sqrt{5(3) + 4} = 3 - \sqrt{15 + 4} = 3 - \sqrt{19}$$ $$\sqrt{3 + 3} - 2 = \sqrt{6} - 2$$ Since direct substitution does not give $0/0$, compute the value: $$\frac{3 - \sqrt{19}}{\sqrt{6} - 2}$$ Step 2: Rationalize the denominator: $$\frac{3 - \sqrt{19}}{\sqrt{6} - 2} \times \frac{\sqrt{6} + 2}{\sqrt{6} + 2} = \frac{(3 - \sqrt{19})(\sqrt{6} + 2)}{6 - 4} = \frac{(3 - \sqrt{19})(\sqrt{6} + 2)}{2}$$ Step 3: Expand numerator: $$3\sqrt{6} + 6 - \sqrt{19}\sqrt{6} - 2\sqrt{19}$$ Final limit: $$\frac{3\sqrt{6} + 6 - \sqrt{114} - 2\sqrt{19}}{2}$$ --- **Problem 2:** Evaluate $$\lim_{x \to 5} \frac{x - x - \sqrt{x + 1}}{\lim_{x \to 1} \frac{-x + 1 - \sqrt{x + 1} + 3}{x - 1}}$$ This expression is ambiguous; assuming the inner limit is: $$\lim_{x \to 1} \frac{-x + 1 - \sqrt{x + 1} + 3}{x - 1}$$ Step 1: Simplify numerator: $$-x + 1 - \sqrt{x + 1} + 3 = 4 - x - \sqrt{x + 1}$$ Step 2: Evaluate the limit as $x \to 1$: Direct substitution gives numerator: $$4 - 1 - \sqrt{2} = 3 - \sqrt{2}$$ Denominator: $$1 - 1 = 0$$ Since denominator is zero, check if numerator also tends to zero: At $x=1$, numerator is $3 - \sqrt{2} \neq 0$, so limit tends to infinity or does not exist. Therefore, the entire expression is undefined or infinite. --- **Problem 3:** Evaluate $$\lim_{x \to 1} \frac{x - \sqrt{x + 4} - 2}{1/x - 2x + 1}$$ Step 1: Substitute $x=1$: Numerator: $$1 - \sqrt{1 + 4} - 2 = 1 - \sqrt{5} - 2 = -1 - \sqrt{5}$$ Denominator: $$1/1 - 2(1) + 1 = 1 - 2 + 1 = 0$$ Since denominator is zero and numerator is not zero, limit tends to infinity or does not exist. --- **Problem 4:** Given function $$g(x) = 2x - \sqrt{x^2 + 2}$$ Evaluate $g(x)$ for given $x$ or analyze behavior. --- **Problem 5:** Evaluate $$\lim_{x \to 0} [f(x) + 3x]$$ where $f(x)$ is not explicitly given. Cannot evaluate without $f(x)$ definition. --- **Summary:** - Problem 1 limit evaluated with rationalization. - Problems 2 and 3 limits tend to infinity or undefined. - Problem 4 function given. - Problem 5 insufficient data. --- "slug": "limit evaluations", "subject": "calculus", "desmos": {"latex": "y=2x - \sqrt{x^2 + 2}", "features": {"intercepts": true, "extrema": true}}, "q_count": 3 Final answer for Problem 1: $$\lim_{x \to 3} \frac{3 - \sqrt{5x + 4}}{\sqrt{x + 3} - 2} = \frac{3\sqrt{6} + 6 - \sqrt{114} - 2\sqrt{19}}{2}$$